In this paper, we study simple splines on a Riemannian manifold $Q$ from the point of view of the Pontryagin maximum principle (PMP) in optimal control theory. The control problem consists in finding smooth curves matching two given tangent vectors with the control being the curve's acceleration, while minimizing a given cost functional. We focus on cubic splines (quadratic cost function) and on time-minimal splines (constant cost function) under bounded acceleration. We present a general strategy to solve for the optimal hamiltonian within the PMP framework based on splitting the variables by means of a linear connection. We write down the corresponding hamiltonian equations in intrinsic form and study the corresponding hamiltonian dynamics in the case $Q$ is the $2$-sphere. We also elaborate on possible applications, including landmark cometrics in computational anatomy.

M. Barbero-Liñán,
Characterization of accessibility for affine connection control systems at some points with nonzero velocity, in Proceedings of the IEEE Conference on Decision and Control and European Control Conference, (2011), 6528-6533.
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M. Barbero-Liñán,
Characterization of accessibility for affine connection control systems at some points with nonzero velocity, in Proceedings of the IEEE Conference on Decision and Control and European Control Conference, (2011), 6528-6533.
Google Scholar

Figure 8.
Top: reconstructed trajectory in the physical sphere, that approaches a neighborhood of an equator. Below: the reduced trajectory emanating from the unstable equilibrium, projected in the $ (v,a)$ plane. Note that $v$ is growing quadratically with respect to $a$. The reconstructed trajectory is approaching a neighborhood of an equator. It remains to be seen if it stays there or returns to a vicinity of the reduced equilibrium.